Antedeluvian - your link does not seem to work - it's correct but got mixed up with the surrounding text and just redirects you to this page. I find it works better if on a separate line:
http://www.amazon.com/Decoding-Heavens-000-Year-Old-Computer---Century-long/dp/0306818612/ref=sr_1_1?ie=UTF8&qid=1371842355&sr=8-1&keywords=Antikythera

I think there are quite a few skill sets that were common in our ancestors' day that have now all but vanished. A lot of them do seem pretty amazing.
Imagine what it took for Lewis and Clark to navigate across the continent without any electronic navigation aids. Who could do that now? How about planing enough food to last for an entire year?
Even looking to more recent times - it wasn't all that long ago when car ownership require enough knowledge to perform a slew of basic repairs, often out away from home.
It might seem impossible now to imagine being able to detect such subtle changes as slowly shortening days, but when that's your life, I suppose you have to be sensitive enough to do that.

Regarding the gradual lengthening and shortening of a day, humans, like cats, can use their stomachs as pretty accurate clocks. Sundials are even more accurate, and can be discovered watching the shadow cast by a tree on the ground. I remember a bit in Lord Dunsany's "Don Rodriguez" where his servant knew that if you wanted to get a good siesta in the shade of a tree, you should lie where the shadow will move to over the next hour or two, rather than lying in the middle of the tree's shadow.
You're also more aware of time if you rise at dawn and go to bed at night-fall, which people did before electric lights and fire. With less distraction from MyTwitFace, you can be more aware of your natural environment in general. If you're not surrounded by tall buildings, it's pretty easy to see that the sun rises and sets at different points on the horizon as the year progresses, provided that you have fixed objects like mountains and trees to compare to.
There were also a whole lot more stars visible in ancient times. Now we have severe light pollution. Anyone seen the Milky Way lately?

These old mechanisms really interest me. Would you recommend this book?
In fact I commented on this calculator in the introduction to my book "How Computers Do Math" in which I said the following:
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Every now and then, strange and wonderful mechanisms from antiquity are discovered. In 1900, for example, a device of unknown purpose containing numerous gear wheels forming a sophisticated mechanism dating from 2200 BC was discovered in a shipwreck close to the tiny Greek island of Antikythera. This contraption, which is now known as the "Antikythera Mechanism" or the "Antikythera Calculator," was created during the early years of the Hellenistic Period: a golden age when science and art flourished in ancient Greece.
In many cases, objects like this prompt speculation that our antediluvian ancestors were the creators of complex mechanical calculators with which they could perform mathematical operations such as addition, subtraction, multiplication, and division. In reality, however, the concept of zero (‘0’) as representing a true quantity in a place-value number system didn’t appear until around 600 AD in India. Without the notion of zero in this context, it is really not possible to create a mechanical calculator in any form we would recognize.
This is not to say that these ancient mechanisms were not incredibly cunning and refined. However, such instruments were probably designed to measure things or to track time in one way or another; for example, to help in predicting the seasons and the activities of celestial objects like the sun, moon, planets, and constellations.

Down under it is the shortest day of the year, and don't we know it (minus 4 degrees C this morning....that's plus 24.8 F for you philistines... though I must admit that does not make it sound so cold...)
But getting back to your point Max I too am in awe of the ancients who discovered things like pi and why we need leap years and how to navigate by the stars, using only their minds.

In conjunction with unveiling of EE Times’ Silicon 60 list, journalist & Silicon 60 researcher Peter Clarke hosts a conversation on startups in the electronics industry. One of Silicon Valley's great contributions to the world has been the demonstration of how the application of entrepreneurship and venture capital to electronics and semiconductor hardware can create wealth with developments in semiconductors, displays, design automation, MEMS and across the breadth of hardware developments. But in recent years concerns have been raised that traditional venture capital has turned its back on hardware-related startups in favor of software and Internet applications and services. Panelists from incubators join Peter Clarke in debate.